Several shape similarity measures, based on shape skeletons, are designed in the context of graph kernels. State-of-the-art kernels act on bags of walks, paths or trails which decompose the skeleton graph, and take into account structural noise through edition mechanisms. However, these approaches fail to capture the complexity of junctions inside skeleton graphs due to the linearity of the pat...

Abstract. Let T be a plane rooted tree with n nodes which is regarded as family tree of a Galton-Watson branching process conditioned on the total progeny. The profile of the tree may be described by the number of nodes or the number of leaves in layer t √ n, respectively. It is shown that these two processes converge weakly to Brownian excursion local time. This is done via characteristic func...

Let P and S be two disjoint sets of n and m points in the plane, respectively. We consider the problem of computing a Steiner tree whose Steiner vertices belong to S, in which each point of P is a leaf, and whose longest edge length is minimum. We present an algorithm that computes such a tree in O((n +m) logm) time, improving the previously best result by a logarithmic factor. We also prove a ...

Let G be a set of disjoint bi-chromatic straight line segments and H be a set of red and blue points in the plane, no three points are collinear. We give tight upper bounds on the maximum degree of a node in the color conforming minimum weight spanning tree (MST) formed by G and H. We also consider bounds on the total length of the edges of 1) the planar MST and the unrestricted MST, 2) the gre...

The Euclidean Steiner tree problem is solved by finding the tree with minimal Euclidean length spanning a set of fixed vertices in the plane, while allowing for the addition of auxiliary vertices (Steiner vertices). Steiner trees are widely used to design real-world structures like highways and oil pipelines. Unfortunately, the Euclidean Steiner Tree Problem has shown to be NP-Hard, meaning the...

In the minimum-cost k-hop spanning tree (k-hop MST) problem, we are given a set S of n points in a metric space, a positive small integer k and a root point r ∈ S. We are interested in computing a rooted spanning tree of minimum cost such that the longest root-leaf path in the tree has at most k edges. We present a polynomial-time approximation scheme for the plane. Our algorithm is based on Ar...

We find a simple, closed formula for the proportion of vertices which are k -protected in all unlabeled rooted plane trees on n vertices. We also find that, as n goes to infinity, the average rank of a random vertex in a tree of size n approaches 0.727649, and the average rank of the root of a tree of size n approaches 1.62297. Mathematics Subject Classification 05A15 · 05A16 · 05C05 tree, enum...

A minimum Steiner tree for a given set X of points is a network interconnecting the points of X having minimum possible total length. The Steiner ratio for a metric space is the largest lower bound for the ratio of lengths between a minimum Steiner tree and a minimum spanning tree on the same set of points in the ,metric space. In this note, we show that for any Minkowski plane, the Steiner rat...

The traditional sona drawing art of Angola is compared from the structural point of view to similar geometric arts found in other parts of the world. A simple algorithm is proposed for constructing a sub-class of sona drawings, called perfect monolinear sona drawings, on a given set of points in the plane, that exhibit the topological structure of a tree. The application of the tree sona drawin...

Given any infinite tree in the plane satisfying certain topological conditions, we construct an entire function f with only two critical values ±1 and no asymptotic such that f−1([−1, 1]) is ambiently homeomorphic to given tree. This can be viewed as a generalization of result Grothendieck (see Schneps (1994)) case trees. Moreover, similar idea leads new proof Nevanlinna (1932) Elfving (1934).